English
If i is an open embedding and f is a morphism, the preimage under i of the range of f is extremally disconnected.
Русский
Если i – открытое вложение и f – морфизм, то прообраз i^{-1}(range f) является экстремально разобщенным.
LaTeX
$$$\operatorname{ExtremallyDisconnected}(i^{-1}(\operatorname{range} f))$$$
Lean4
theorem extremallyDisconnected_preimage : ExtremallyDisconnected (i ⁻¹' (Set.range f)) where
open_closure U
hU :=
by
have h : IsClopen (i ⁻¹' (Set.range f)) :=
⟨IsClosed.preimage i.hom.continuous (isCompact_range f.hom.continuous).isClosed,
IsOpen.preimage i.hom.continuous hi.isOpen_range⟩
rw [← (closure U).preimage_image_eq Subtype.coe_injective, ← h.1.isClosedEmbedding_subtypeVal.closure_image_eq U]
exact isOpen_induced (ExtremallyDisconnected.open_closure _ (h.2.isOpenEmbedding_subtypeVal.isOpenMap U hU))
